libs/math/doc/html/math_toolkit/dist_ref/dists/nc_beta_dist.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Noncentral Beta Distribution</title>
   5 <link rel="stylesheet" href="../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../dists.html" title="Distributions">
   9 <link rel="prev" href="negative_binomial_dist.html" title="Negative Binomial Distribution">
  10 <link rel="next" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="negative_binomial_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_chi_squared_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.nc_beta_dist"></a><a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
  28         Beta Distribution</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">non_central_beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">non_central_beta_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_beta</span><span class="special">;</span>
  38
  39 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  40 <span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  45
  46    <span class="comment">// Constructor:</span>
  47    <span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  48
  49    <span class="comment">// Accessor to shape parameters:</span>
  50    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  51    <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  52
  53    <span class="comment">// Accessor to non-centrality parameter lambda:</span>
  54    <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  55 <span class="special">};</span>
  56
  57 <span class="special">}}</span> <span class="comment">// namespaces</span>
  58 </pre>
  59 <p>
  60           The noncentral beta distribution is a generalization of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta
  61           Distribution</a>.
  62         </p>
  63 <p>
  64           It is defined as the ratio
  65         </p>
  66 <div class="blockquote"><blockquote class="blockquote"><p>
  67             <span class="serif_italic">X = &#967;<sub>m</sub><sup>2</sup>(&#955;) / (&#967;<sub>m</sub><sup>2</sup>(&#955;) + &#967;<sub>n</sub><sup>2</sup>)</span>
  68           </p></blockquote></div>
  69 <p>
  70           where <span class="serif_italic">&#967;<sub>m</sub><sup>2</sup>(&#955;)</span> is a noncentral <span class="serif_italic">&#967;<sup>2</sup></span> random variable with <span class="emphasis"><em>m</em></span>
  71           degrees of freedom, and &#967;<sub>n</sub><sup>2</sup>
  72 is a central <span class="serif_italic">&#967;<sup>2</sup> </span>
  73           random variable with <span class="emphasis"><em>n</em></span> degrees of freedom.
  74         </p>
  75 <p>
  76           This gives a PDF that can be expressed as a Poisson mixture of beta distribution
  77           PDFs:
  78         </p>
  79 <div class="blockquote"><blockquote class="blockquote"><p>
  80             <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
  81
  82           </p></blockquote></div>
  83 <p>
  84           where P(i;&#955;/2) is the discrete Poisson probablity at <span class="emphasis"><em>i</em></span>,
  85           with mean &#955;/2, and I<sub>x</sub><sup>'</sup>(&#945;, &#946;) is the derivative of the incomplete beta function.
  86           This leads to the usual form of the CDF as:
  87         </p>
  88 <div class="blockquote"><blockquote class="blockquote"><p>
  89             <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span>
  90
  91           </p></blockquote></div>
  92 <p>
  93           The following graph illustrates how the distribution changes for different
  94           values of &#955;:
  95         </p>
  96 <div class="blockquote"><blockquote class="blockquote"><p>
  97             <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span>
  98
  99           </p></blockquote></div>
 100 <h5>
 101 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a>
 102           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.member_functions"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.member_functions">Member
 103           Functions</a>
 104         </h5>
 105 <pre class="programlisting"><span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
 106 </pre>
 107 <p>
 108           Constructs a noncentral beta distribution with shape parameters <span class="emphasis"><em>a</em></span>
 109           and <span class="emphasis"><em>b</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
 110         </p>
 111 <p>
 112           Requires a &gt; 0, b &gt; 0 and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 113         </p>
 114 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 115 </pre>
 116 <p>
 117           Returns the parameter <span class="emphasis"><em>a</em></span> from which this object was
 118           constructed.
 119         </p>
 120 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 121 </pre>
 122 <p>
 123           Returns the parameter <span class="emphasis"><em>b</em></span> from which this object was
 124           constructed.
 125         </p>
 126 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 127 </pre>
 128 <p>
 129           Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object
 130           was constructed.
 131         </p>
 132 <h5>
 133 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h1"></a>
 134           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors">Non-member
 135           Accessors</a>
 136         </h5>
 137 <p>
 138           Most of the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
 139           accessor functions</a> are supported: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 140           Distribution Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
 141           Density Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>,
 142           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 143           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 144           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
 145           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>,
 146           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 147           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 148         </p>
 149 <p>
 150           Mean and variance are implemented using hypergeometric pfq functions and
 151           relations given in <a href="http://reference.wolfram.com/mathematica/ref/NoncentralBetaDistribution.html" target="_top">Wolfram
 152           Noncentral Beta Distribution</a>.
 153         </p>
 154 <p>
 155           However, the following are not currently implemented: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>,
 156           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and
 157           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>.
 158         </p>
 159 <p>
 160           The domain of the random variable is [0, 1].
 161         </p>
 162 <h5>
 163 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h2"></a>
 164           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.accuracy"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.accuracy">Accuracy</a>
 165         </h5>
 166 <p>
 167           The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>)
 168           found on various platforms with various floating point types. The failures
 169           in the comparison to the <a href="http://www.r-project.org/" target="_top">R Math
 170           library</a>, seem to be mostly in the corner cases when the probablity
 171           would be very small. Unless otherwise specified any floating-point type
 172           that is narrower than the one shown will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 173           zero error</a>.
 174         </p>
 175 <div class="table">
 176 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF"></a><p class="title"><b>Table&#160;5.4.&#160;Error rates for non central beta CDF</b></p>
 177 <div class="table-contents"><table class="table" summary="Error rates for non central beta CDF">
 178 <colgroup>
 179 <col>
 180 <col>
 181 <col>
 182 <col>
 183 <col>
 184 </colgroup>
 185 <thead><tr>
 186 <th>
 187                 </th>
 188 <th>
 189                   <p>
 190                     GNU C++ version 7.1.0<br> linux<br> double
 191                   </p>
 192                 </th>
 193 <th>
 194                   <p>
 195                     GNU C++ version 7.1.0<br> linux<br> long double
 196                   </p>
 197                 </th>
 198 <th>
 199                   <p>
 200                     Sun compiler version 0x5150<br> Sun Solaris<br> long double
 201                   </p>
 202                 </th>
 203 <th>
 204                   <p>
 205                     Microsoft Visual C++ version 14.1<br> Win32<br> double
 206                   </p>
 207                 </th>
 208 </tr></thead>
 209 <tbody>
 210 <tr>
 211 <td>
 212                   <p>
 213                     Non Central Beta, medium parameters
 214                   </p>
 215                 </td>
 216 <td>
 217                   <p>
 218                     <span class="blue">Max = 0.998&#949; (Mean = 0.0649&#949;)</span><br>
 219                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 220                     = 1.46e+26&#949; (Mean = 3.5e+24&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_Rmath_3_2_3_Non_Central_Beta_medium_parameters">And
 221                     other failures.</a>)</span>
 222                   </p>
 223                 </td>
 224 <td>
 225                   <p>
 226                     <span class="blue">Max = 824&#949; (Mean = 27.4&#949;)</span>
 227                   </p>
 228                 </td>
 229 <td>
 230                   <p>
 231                     <span class="blue">Max = 832&#949; (Mean = 38.1&#949;)</span>
 232                   </p>
 233                 </td>
 234 <td>
 235                   <p>
 236                     <span class="blue">Max = 242&#949; (Mean = 31&#949;)</span>
 237                   </p>
 238                 </td>
 239 </tr>
 240 <tr>
 241 <td>
 242                   <p>
 243                     Non Central Beta, large parameters
 244                   </p>
 245                 </td>
 246 <td>
 247                   <p>
 248                     <span class="blue">Max = 1.18&#949; (Mean = 0.175&#949;)</span><br>
 249                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 250                     = 1.01e+36&#949; (Mean = 1.19e+35&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_Rmath_3_2_3_Non_Central_Beta_large_parameters">And
 251                     other failures.</a>)</span>
 252                   </p>
 253                 </td>
 254 <td>
 255                   <p>
 256                     <span class="blue">Max = 2.5e+04&#949; (Mean = 3.78e+03&#949;)</span>
 257                   </p>
 258                 </td>
 259 <td>
 260                   <p>
 261                     <span class="blue">Max = 2.57e+04&#949; (Mean = 4.45e+03&#949;)</span>
 262                   </p>
 263                 </td>
 264 <td>
 265                   <p>
 266                     <span class="blue">Max = 3.66e+03&#949; (Mean = 500&#949;)</span>
 267                   </p>
 268                 </td>
 269 </tr>
 270 </tbody>
 271 </table></div>
 272 </div>
 273 <br class="table-break"><div class="table">
 274 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF_complement"></a><p class="title"><b>Table&#160;5.5.&#160;Error rates for non central beta CDF complement</b></p>
 275 <div class="table-contents"><table class="table" summary="Error rates for non central beta CDF complement">
 276 <colgroup>
 277 <col>
 278 <col>
 279 <col>
 280 <col>
 281 <col>
 282 </colgroup>
 283 <thead><tr>
 284 <th>
 285                 </th>
 286 <th>
 287                   <p>
 288                     GNU C++ version 7.1.0<br> linux<br> double
 289                   </p>
 290                 </th>
 291 <th>
 292                   <p>
 293                     GNU C++ version 7.1.0<br> linux<br> long double
 294                   </p>
 295                 </th>
 296 <th>
 297                   <p>
 298                     Sun compiler version 0x5150<br> Sun Solaris<br> long double
 299                   </p>
 300                 </th>
 301 <th>
 302                   <p>
 303                     Microsoft Visual C++ version 14.1<br> Win32<br> double
 304                   </p>
 305                 </th>
 306 </tr></thead>
 307 <tbody>
 308 <tr>
 309 <td>
 310                   <p>
 311                     Non Central Beta, medium parameters
 312                   </p>
 313                 </td>
 314 <td>
 315                   <p>
 316                     <span class="blue">Max = 0.998&#949; (Mean = 0.0936&#949;)</span><br>
 317                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 318                     = 7.5e+97&#949; (Mean = 1.37e+96&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_2_3_Non_Central_Beta_medium_parameters">And
 319                     other failures.</a>)</span>
 320                   </p>
 321                 </td>
 322 <td>
 323                   <p>
 324                     <span class="blue">Max = 396&#949; (Mean = 50.7&#949;)</span>
 325                   </p>
 326                 </td>
 327 <td>
 328                   <p>
 329                     <span class="blue">Max = 554&#949; (Mean = 57.2&#949;)</span>
 330                   </p>
 331                 </td>
 332 <td>
 333                   <p>
 334                     <span class="blue">Max = 624&#949; (Mean = 62.7&#949;)</span>
 335                   </p>
 336                 </td>
 337 </tr>
 338 <tr>
 339 <td>
 340                   <p>
 341                     Non Central Beta, large parameters
 342                   </p>
 343                 </td>
 344 <td>
 345                   <p>
 346                     <span class="blue">Max = 0.986&#949; (Mean = 0.188&#949;)</span><br>
 347                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 348                     = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_2_3_Non_Central_Beta_large_parameters">And
 349                     other failures.</a>)</span>
 350                   </p>
 351                 </td>
 352 <td>
 353                   <p>
 354                     <span class="blue">Max = 6.83e+03&#949; (Mean = 993&#949;)</span>
 355                   </p>
 356                 </td>
 357 <td>
 358                   <p>
 359                     <span class="blue">Max = 3.56e+03&#949; (Mean = 707&#949;)</span>
 360                   </p>
 361                 </td>
 362 <td>
 363                   <p>
 364                     <span class="blue">Max = 1.25e+04&#949; (Mean = 1.49e+03&#949;)</span>
 365                   </p>
 366                 </td>
 367 </tr>
 368 </tbody>
 369 </table></div>
 370 </div>
 371 <br class="table-break"><p>
 372           Error rates for the PDF, the complement of the CDF and for the quantile
 373           functions are broadly similar.
 374         </p>
 375 <h5>
 376 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h3"></a>
 377           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.tests"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.tests">Tests</a>
 378         </h5>
 379 <p>
 380           There are two sets of test data used to verify this implementation: firstly
 381           we can compare with a few sample values generated by the <a href="http://www.r-project.org/" target="_top">R
 382           library</a>. Secondly, we have tables of test data, computed with this
 383           implementation and using interval arithmetic - this data should be accurate
 384           to at least 50 decimal digits - and is the used for our accuracy tests.
 385         </p>
 386 <h5>
 387 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h4"></a>
 388           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.implementation"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.implementation">Implementation</a>
 389         </h5>
 390 <p>
 391           The CDF and its complement are evaluated as follows:
 392         </p>
 393 <p>
 394           First we determine which of the two values (the CDF or its complement)
 395           is likely to be the smaller, the crossover point is taken to be the mean
 396           of the distribution: for this we use the approximation due to: R. Chattamvelli
 397           and R. Shanmugam, "Algorithm AS 310: Computing the Non-Central Beta
 398           Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997),
 399           pp. 146-156.
 400         </p>
 401 <div class="blockquote"><blockquote class="blockquote"><p>
 402             <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span>
 403
 404           </p></blockquote></div>
 405 <p>
 406           Then either the CDF or its complement is computed using the relations:
 407         </p>
 408 <div class="blockquote"><blockquote class="blockquote"><p>
 409             <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span>
 410
 411           </p></blockquote></div>
 412 <p>
 413           The summation is performed by starting at i = &#955;/2, and then recursing in
 414           both directions, using the usual recurrence relations for the Poisson PDF
 415           and incomplete beta functions. This is the "Method 2" described
 416           by:
 417         </p>
 418 <p>
 419           Denise Benton and K. Krishnamoorthy, "Computing discrete mixtures
 420           of continuous distributions: noncentral chisquare, noncentral t and the
 421           distribution of the square of the sample multiple correlation coefficient",
 422           Computational Statistics &amp; Data Analysis 43 (2003) 249-267.
 423         </p>
 424 <p>
 425           Specific applications of the above formulae to the noncentral beta distribution
 426           can be found in:
 427         </p>
 428 <p>
 429           Russell V. Lenth, "Algorithm AS 226: Computing Noncentral Beta Probabilities",
 430           Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244.
 431         </p>
 432 <p>
 433           H. Frick, "Algorithm AS R84: A Remark on Algorithm AS 226: Computing
 434           Non-Central Beta Probabilities", Applied Statistics, Vol. 39, No.
 435           2. (1990), pp. 311-312.
 436         </p>
 437 <p>
 438           Ming Long Lam, "Remark AS R95: A Remark on Algorithm AS 226: Computing
 439           Non-Central Beta Probabilities", Applied Statistics, Vol. 44, No.
 440           4. (1995), pp. 551-552.
 441         </p>
 442 <p>
 443           Harry O. Posten, "An Effective Algorithm for the Noncentral Beta Distribution
 444           Function", The American Statistician, Vol. 47, No. 2. (May, 1993),
 445           pp. 129-131.
 446         </p>
 447 <p>
 448           R. Chattamvelli, "A Note on the Noncentral Beta Distribution Function",
 449           The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234.
 450         </p>
 451 <p>
 452           Of these, the Posten reference provides the most complete overview, and
 453           includes the modification starting iteration at &#955;/2.
 454         </p>
 455 <p>
 456           The main difference between this implementation and the above references
 457           is the direct computation of the complement when most efficient to do so,
 458           and the accumulation of the sum to -1 rather than subtracting the result
 459           from 1 at the end: this can substantially reduce the number of iterations
 460           required when the result is near 1.
 461         </p>
 462 <p>
 463           The PDF is computed using the methodology of Benton and Krishnamoorthy
 464           and the relation:
 465         </p>
 466 <div class="blockquote"><blockquote class="blockquote"><p>
 467             <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
 468
 469           </p></blockquote></div>
 470 <p>
 471           Quantiles are computed using a specially modified version of <a class="link" href="../../roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket
 472           and solve</a>, starting the search for the root at the mean of the distribution.
 473           (A Cornish-Fisher type expansion was also tried, but while this gets quite
 474           close to the root in many cases, when it is wrong it tends to introduce
 475           quite pathological behaviour: more investigation in this area is probably
 476           warranted).
 477         </p>
 478 </div>
 479 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 480 <td align="left"></td>
 481 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 482       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 483       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 484       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 485       Daryle Walker and Xiaogang Zhang<p>
 486         Distributed under the Boost Software License, Version 1.0. (See accompanying
 487         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 488       </p>
 489 </div></td>
 490 </tr></table>
 491 <hr>
 492 <div class="spirit-nav">
 493 <a accesskey="p" href="negative_binomial_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_chi_squared_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
 494 </div>
 495 </body>
 496 </html>